Properties

Label 170.2.a.b
Level $170$
Weight $2$
Character orbit 170.a
Self dual yes
Analytic conductor $1.357$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.35745683436\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - 2q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - 2q^{7} - q^{8} + q^{9} - q^{10} - 2q^{11} - 2q^{12} - 6q^{13} + 2q^{14} - 2q^{15} + q^{16} + q^{17} - q^{18} - 8q^{19} + q^{20} + 4q^{21} + 2q^{22} - 2q^{23} + 2q^{24} + q^{25} + 6q^{26} + 4q^{27} - 2q^{28} + 6q^{29} + 2q^{30} - 2q^{31} - q^{32} + 4q^{33} - q^{34} - 2q^{35} + q^{36} + 6q^{37} + 8q^{38} + 12q^{39} - q^{40} + 2q^{41} - 4q^{42} - 4q^{43} - 2q^{44} + q^{45} + 2q^{46} + 4q^{47} - 2q^{48} - 3q^{49} - q^{50} - 2q^{51} - 6q^{52} - 10q^{53} - 4q^{54} - 2q^{55} + 2q^{56} + 16q^{57} - 6q^{58} - 2q^{60} - 10q^{61} + 2q^{62} - 2q^{63} + q^{64} - 6q^{65} - 4q^{66} + 8q^{67} + q^{68} + 4q^{69} + 2q^{70} + 14q^{71} - q^{72} + 10q^{73} - 6q^{74} - 2q^{75} - 8q^{76} + 4q^{77} - 12q^{78} - 14q^{79} + q^{80} - 11q^{81} - 2q^{82} - 4q^{83} + 4q^{84} + q^{85} + 4q^{86} - 12q^{87} + 2q^{88} + 6q^{89} - q^{90} + 12q^{91} - 2q^{92} + 4q^{93} - 4q^{94} - 8q^{95} + 2q^{96} - 14q^{97} + 3q^{98} - 2q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 −2.00000 1.00000 1.00000 2.00000 −2.00000 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(17\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 170.2.a.b 1
3.b odd 2 1 1530.2.a.j 1
4.b odd 2 1 1360.2.a.j 1
5.b even 2 1 850.2.a.k 1
5.c odd 4 2 850.2.c.f 2
7.b odd 2 1 8330.2.a.l 1
8.b even 2 1 5440.2.a.u 1
8.d odd 2 1 5440.2.a.c 1
15.d odd 2 1 7650.2.a.bc 1
17.b even 2 1 2890.2.a.h 1
17.c even 4 2 2890.2.b.e 2
20.d odd 2 1 6800.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.a.b 1 1.a even 1 1 trivial
850.2.a.k 1 5.b even 2 1
850.2.c.f 2 5.c odd 4 2
1360.2.a.j 1 4.b odd 2 1
1530.2.a.j 1 3.b odd 2 1
2890.2.a.h 1 17.b even 2 1
2890.2.b.e 2 17.c even 4 2
5440.2.a.c 1 8.d odd 2 1
5440.2.a.u 1 8.b even 2 1
6800.2.a.c 1 20.d odd 2 1
7650.2.a.bc 1 15.d odd 2 1
8330.2.a.l 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(170))\):

\( T_{3} + 2 \)
\( T_{7} + 2 \)
\( T_{13} + 6 \)